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Asymptotic Inference for Nonstationary Fractionally Integrated Processes

  • Francesc Marmol

    ()

    (Universidad Carlos III)

  • Juan J. Dolado

    ()

    (Universidad Carlos III)

This paper studies the asymptotics of nonstationary fractionally integrated (NFI) multivariate processes with memory parameter d > 0.5 . We provide conditions to establish a functional central limit theorem and weak convergence of stochastic integrals for NFI processes under the assumption that the innovations are linear processes. Several applications of these results are given. More specifically, we prove the rates of convergence of the OLS estimators of cointegrating vectors in triangular representations. When the regressors are strongly exogenous with respect to the corresponding parameters of the cointegrating vector, then the limiting OLS distribution is a mixture of normals from which standard inference can be implemented. On the other hand, we also extend Sims, Stock and Watson's (1990) study on estimation and hypothesis testing in vector autoregressions with integrated processes and deterministic components to the more general fractional framework. We show how their main conclusions remain valid when dealing with NFI processes, namely, that whenever a block of coefficients can be written as coefficients on zero mean I(0) regressors in a model that includes a constant term, they will have a joint asymptotic normal distribution, so that the corresponding restrictions can be tested using standard asymptotic chi-squared distribution theory. Otherwise, in general, the associated statistics will have nonstandard limiting distributions.

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Paper provided by Society for Computational Economics in its series Computing in Economics and Finance 1999 with number 513.

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Date of creation: 01 Mar 1999
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Handle: RePEc:sce:scecf9:513
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  1. Park, Joon Y. & Phillips, Peter C.B., 1989. "Statistical Inference in Regressions with Integrated Processes: Part 2," Econometric Theory, Cambridge University Press, vol. 5(01), pages 95-131, April.
  2. Liu, Ming, 1998. "Asymptotics Of Nonstationary Fractional Integrated Series," Econometric Theory, Cambridge University Press, vol. 14(05), pages 641-662, October.
  3. James H. Stock & Mark W. Watson, 1991. "A simple estimator of cointegrating vectors in higher order integrated systems," Working Paper Series, Macroeconomic Issues 91-3, Federal Reserve Bank of Chicago.
  4. Toda, Hiro Y & Phillips, Peter C B, 1993. "Vector Autoregressions and Causality," Econometrica, Econometric Society, vol. 61(6), pages 1367-93, November.
  5. Baillie, Richard T., 1996. "Long memory processes and fractional integration in econometrics," Journal of Econometrics, Elsevier, vol. 73(1), pages 5-59, July.
  6. Jeganathan, P., 1999. "On Asymptotic Inference In Cointegrated Time Series With Fractionally Integrated Errors," Econometric Theory, Cambridge University Press, vol. 15(04), pages 583-621, August.
  7. Peter C.B. Phillips & Joon Y. Park, 1986. "Statistical Inference in Regressions with Integrated Processes: Part 1," Cowles Foundation Discussion Papers 811R, Cowles Foundation for Research in Economics, Yale University, revised Aug 1987.
  8. Stock, James H, 1987. "Asymptotic Properties of Least Squares Estimators of Cointegrating Vectors," Econometrica, Econometric Society, vol. 55(5), pages 1035-56, September.
  9. Sims, Christopher A & Stock, James H & Watson, Mark W, 1990. "Inference in Linear Time Series Models with Some Unit Roots," Econometrica, Econometric Society, vol. 58(1), pages 113-44, January.
  10. Peter C.B. Phillips, 1987. "Weak Convergence of Sample Covariance Matrices to Stochastic Integrals via Martingale Approximations," Cowles Foundation Discussion Papers 846, Cowles Foundation for Research in Economics, Yale University.
  11. D. Marinucci, 1998. "Band spectrum regression for cointegrated time series with long memory innovations," LSE Research Online Documents on Economics 6871, London School of Economics and Political Science, LSE Library.
  12. D Marinucci & Peter M. Robinson, 1998. "Weak convergence of multivariate fractional processes," LSE Research Online Documents on Economics 2322, London School of Economics and Political Science, LSE Library.
  13. D Marinucci, 1998. "Band Spectrum Regression for Cointegrated Time Series with Long Memory Innovations," STICERD - Econometrics Paper Series 353, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
  14. P. C. B. Phillips & S. N. Durlauf, 1986. "Multiple Time Series Regression with Integrated Processes," Review of Economic Studies, Oxford University Press, vol. 53(4), pages 473-495.
  15. D Marinucci & Peter M. Robinson, 1998. "Semiparametric frequency domain analysis of fractional cointegration," LSE Research Online Documents on Economics 2258, London School of Economics and Political Science, LSE Library.
  16. Hansen, Bruce E., 1992. "Convergence to Stochastic Integrals for Dependent Heterogeneous Processes," Econometric Theory, Cambridge University Press, vol. 8(04), pages 489-500, December.
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